Spring 2024
Time & Location: All talks are on Tuesdays in Norman Mayer G106 at 3:30PM unless otherwise noted.
Organizer: Sang-Eun Lee
September 10
Title: Transition of Equilibria in the Dynamics of Chemically Active Particles
Sang-Eun Lee - Tulane University
Abstract: Motivated by autophoretic droplet swimmers, we present a dynamical transition to the chemically active swimmers. With the periodic boundary condition imposed to the reaction-diffusion system, the particles secrete the chemicals with un unlimited supply and avoid the chemical so have anti-chemotactic. Due to the periodic boundary condition, there is an interesting transition of equilibrium of particles due to a dimensionless parameter in both one and two dimensions. In addition, we present an effect of fluid that affects the convection of both chemicals and the particles.
Time: 3:30 pm
Location: Norman Mayer G106
September 10
Title: A Kirchhoff Rod Model to Study Dynamics of Flexible Fibers and Rotating Helical Flagella in Stokes Flow
Rubaiyat Islam - Tulane University
Abstract: At the microscale, fluid motion is governed by Stokes equations. To study how microfibers behave in an ambient flow or how bacteria propel themselves with their rotating helical flagellum, we need a model to compute forces and torques along the fiber body and a way to include fluid interactions. Our computational framework is a Kirchhoff rod model coupled to regularized Stokeslet segments. This model takes advantage of the slenderness of fibers or flagella and uses a set of orthonormal triads to compute forces and torques. Passive filaments show rich shape deformations depending on their length and stiffness when subject to a background flow. Using a system of images for Stokeslet segments, we also show flagellated bacteria swimming in circles near a rigid wall.
Time: 3:30 pm
Location: Norman Mayer G106
September 24
Title: Modular Forms and Orders of Vanishing
Peter Marcus - Tulane University
Abstract: Modular forms are holomorphic functions with specific symmetry properties. Their Fourier expansions are generating functions for various sequences of interest, such as partition numbers and divisor sums. These functions live in finite-dimensional vector spaces, so by studying these spaces we can learn about these number-theoretic sequences. The dimensions of these spaces have well-known formulas, but there is no known formula for the maximal order of vanishing. In other words, if you write a row-reduced basis of Fourier expansions, when will the first nonzero Fourier coefficient of the last function occur? I will give an overview of this subject and progress on this problem.
Time: 3:30 pm
Location: Norman Mayer G106
October 1
Title: Tevelev Degrees of $\mathbb{P}^1$
Naufil Sakran - Tulane University
Abstract: In this talk, I will introduce the field of enumerative geometry and discuss recent developments in this area. Let $C$ be a general curve with $n$ marked points, and consider a degree $d$ map $\pi: C \to \mathbb{P}^1$, subject to specific incidence conditions.
Tevelev degree is defined as the number of such maps $\pi: C \to \mathbb{P}^1$. Interestingly, the computation of Tevelev degree presents intriguing connections with combinatorics, particularly through Dyck path counting and Schubert calculus. I will explore these topics, discussing their role in computing Tevelev degrees and presenting the latest formulas, as featured in recent work by R. Pandharipande, A. Cela, C. Lian, and others.
Time: 3:30 pm
Location: Norman Mayer G106
October 15
Title: Regularized Non-uniform Segments
Zheng Wang - Tulane University
Abstract: Regularized Stokeslet segment is a method used in fluid dynamics to model the motion of slender, flexible filaments (such as biological flagella, cilia, or fibers) immersed in a viscous fluid with low Reynolds numbers. The regularization parameter is usually an approximation of the filament cross-section radius. In this study, we present a modified model where the regularization parameter is not a constant but a continuous function. This method could be used to model filaments who has non-uniform radius.
Time: 3:30 pm
Location: Stanley Thomas 316
October 29
Title: The Darker Side of Mathematics
Nathaniel Vaduthala - Tulane University
Abstract: In this talk, we will discuss the mathematical contributions of history’s most notorious mathematicians, accompanied by a brief biographical overview of their lives.
Time: 3:30 pm
Location: F. Edward Hebert Hall 213
November 5
Title: Interpolation Problem and some Recent Developments
Dipendranath Mahato - Tulane University
Abstract: The classical Interpolation problem is a numerical analysis problem that deals on estimating some new data, from a known set of data. There have been different approaches to provide more refined numerical results on Interpolation, like Polynomial Interpolation, Spline Interpolation, Mimetic Interpolation etc. But in higher dimensions (specifically in $\mathbb{P}^N$), the main problem is to find the minimal degree homogeneous polynomial that vanishes on a finite set of points with given set of multiplicities. To deal such problem G.V. Chudnovsky and J.P. Demailly provided some conjectural bounds to the minimal degree, which I am going to discuss in my talk. I will also talk on some recent developments on this topic.
Time: 3:30 pm
Location: F. Edward Hebert Hall 213
November 12
Title: The unique factorization problem on the rings of integers
Ngoc Trinh Le - Tulane University
Abstract: By the Fundamental Theorem of Arithmetic, we know that the ring of integer numbers ($\mathbb{Z}$) admits the unique factorization into prime integers. However, this property is not retained when we expand our ring $\mathbb{Z}$ (e.g. the ring $\mathbb{Z}[\sqrt{-19}]}$ where $35=5.7=(4+\sqrt{-19})(4-\sqrt{-19}))$. One approach to this problem is generalizing the factorization from the elements into the ideals level. Fortunately, in the cases of ring of integers, this unique factorization property turns out to be true on the latter sense. In this talk, I will give an brief introduction about the unique factorization property on both levels and show how we can use this property to solve some Diophantine equations.
Time: 3:30 pm
Location: F. Edward Hebert Hall 213
November 19
Title: Reed-Solomon Codes in the Uniform Tree Metric
Dillon Montero - Tulane University
Abstract: The classical theory of error-correcting codes primarily uses the Hamming metric to measure distances. Within this framework, Maximum Distance Separable (MDS) codes are highly valued due to their optimal parameters, enabling the correction of the maximum number of errors for a given code rate. In 1997, Rosenbloom and Tsfasman introduced the m-metric codes (now known as NRT metric codes), identifying analogs of Reed-Solomon codes that still possess the MDS property. Later, Skriganov provided an explicit construction of these codes using Hasse derivatives.
In this talk, we will introduce and discuss our analogs of Reed-Solomon codes in a related context, also employing Hasse derivatives for our construction.
This is joint work with Mahir Bilen Can.
Time: 3:30 pm
Location: F. Edward Hebert Hall 213
